Let f: C→C be the function f(2)= wz where @ = u + iv is a fixed complex number. (a) Write down the formula for f viewed as a function f: R² → R², i.e. give formulas for Re(f) and Im(f) in terms of x = Re(z) and y = Im(z). (b) Compute the total derivative Df of fat z as a 2 by 2 matrix with real entries. (c) Describe the linear transformation of the plane determined by this matrix and relate it to the complex number .
Let f: C→C be the function f(2)= wz where @ = u + iv is a fixed complex number. (a) Write down the formula for f viewed as a function f: R² → R², i.e. give formulas for Re(f) and Im(f) in terms of x = Re(z) and y = Im(z). (b) Compute the total derivative Df of fat z as a 2 by 2 matrix with real entries. (c) Describe the linear transformation of the plane determined by this matrix and relate it to the complex number .
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 22E
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![Let f: C → C be the function ƒ(z) = w z where w = u + iv is a fixed complex number.
(a) Write down the formula for f viewed as a function ƒ : R² → R², i.e. give formulas for Re(f)
and Im(f) in terms of x = Re(z) and y = Im(z).
(b) Compute the total derivative Df₂ of ƒ at z as a 2 by 2 matrix with real entries.
(c) Describe the linear transformation of the plane determined by this matrix and relate it to the
complex number w.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0da95474-4618-47f0-9f0f-17505675dd41%2F21bc270f-e04b-43fc-8299-9406f5a12cf7%2Fdf4vcs9_processed.png&w=3840&q=75)
Transcribed Image Text:Let f: C → C be the function ƒ(z) = w z where w = u + iv is a fixed complex number.
(a) Write down the formula for f viewed as a function ƒ : R² → R², i.e. give formulas for Re(f)
and Im(f) in terms of x = Re(z) and y = Im(z).
(b) Compute the total derivative Df₂ of ƒ at z as a 2 by 2 matrix with real entries.
(c) Describe the linear transformation of the plane determined by this matrix and relate it to the
complex number w.
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